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no delay methodology for a mixed-priority crossing situation, Variable Definitions
where drivers sometimes yield to create crossing opportuni-
Many of the variables used in the delay model development
ties but where pedestrians sometimes have to rely on their
are similar to the ones included in the Chapter 5 analysis and
judgment of gaps in traffic to cross the street. The HCM gap-
acceptancebased methods are limited to cases where pedestri- have been defined in Chapter 4. Initial independent variables
ans have full priority (100% of traffic yields) or where drivers included P(Y_ENC), P(CG_ENC), P(GO|Y), and P(GO|CG).
have priority (no yields) and pedestrians are limited to cross- The dependent variable for the models is the average pedes-
ings in gaps between moving vehicles only. An updated pedes- trian delay from the time a trial started until a crossing was
trian delay model that allows for a reduction of pedestrian initiated.
delay due to driver yielding is being considered for the 2010 For model development, some additional explanatory vari-
release of the HCM. However, the proposed theoretical model ables are defined. They are obtained by manipulating the orig-
is not calibrated from field data and does not distinguish inal behavioral probabilities.
between different subpopulations of pedestrians.
· P(Y_and_GO): The probability of crossing in a yield, defined
With currently available HCM pedestrian delay models,
it is therefore not possible to represent the mixed-priority as the probability of encountering a yield multiplied by the
interaction that was observed at the studied CTL and round- probability of utilizing a yield:
about crosswalks. Furthermore, the HCM approach does not P(Y_and_GO) = P(Y_ENC) P(GO|Y).
adequately capture the observed utilization rates of cross- · P(CG_and_GO): The probability of crossing in a crossable
ing opportunities. For example, a gap-acceptancebased gap, defined as the probability of encountering a CG multi-
delay model assumes that pedestrians utilize every cross- plied by the probability of utilizing a CG:
able gap, which was found not to be the case for blind pedes- P(CG_and_GO) = P(CG_ENC) P(GO|CG).
trians. This section develops mixed-priority pedestrian delay · P(Cross): The probability of crossing, defined as the sum of
models that capture the mix of yields and crossable gaps the probabilities of crossing in a yield or crossing in a cross-
encountered and acknowledge the different utilization rates able gap.
observed for different sites and by different participants. A P(Cross) = P(Y_and_GO) + P(CG_and_GO)
more detailed description of the delay model development
is given in Appendix K. Different delay models were developed for the three types
of sites: CTL, single-lane roundabout, and two-lane round-
about. Some additional binary variables were defined to dis-
Approach tinguish between different sites, different crossings, different
The mixed-priority delay models are developed on the prem- treatments, and pretest and posttest treatment periods.
ise of the accessibility framework presented in Chapter 4. It The model development uses a multi-linear regression
is hypothesized that the rates of occurrence and utilization approach to predict the dependent variable, delay, as a func-
of yield and gap crossing opportunities are correlated to tion of various independent variables. All variables are given on
pedestrian delay. Using a multi-linear regression approach, a per-leg basis at the roundabout, and as a result, the total delay
the dependent variable, delay, can therefore be described as for a two-stage crossing at a roundabout is twice the estimate
a function of the four probability parameters P(Y_ENC), (assuming the probabilities are the same). CTL crossings are
P(CG_ENC), P(GO|Y), and P(GO|CG). Chapter 5 contains single-stage only. A histogram of the distribution of the delay
the raw data used to generate the results for each participant at variable showed significant skew to the left, suggesting a log-
each of the test sites, along with the average delay experienced normal distribution. Consequently, all predictive probability
by that participant. variables were transformed by applying the natural logarithm
In the field experiments, blind participants crossed inde- of the variable. All regression is performed in SAS statistical
pendently at three different single-lane roundabouts, two analysis software (SAS 1999) using PROC GLM, a procedure
crosswalks at a two-lane roundabout, and two crosswalks at an to perform multi-linear regression.
intersection with CTLs. In each study, each participant crossed
multiple times. Each trial consisted of four lane crossings at
Results
roundabouts (for example, entryexitexitentry) and two
lane crossings for the CTL (curbisland and islandcurb). At This section presents an overview of the resulting delay mod-
each site, every pedestrian completed multiple trials to obtain els. Results are presented consecutively for the CTL, single-lane
an estimate of average crossing performance. These crossing- roundabout, and two-lane roundabout models, respectively.
specific averages were used in the mixed-priority delay model The results include the recommended delay equation for the
development. three classes of crossings as well as a graphical representation
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of the models. The graphs are only shown for illustrative pur- Equation 1. Suggested pedestrian delay model for CTL.
poses since only one or two dimensions can be shown at one
time. For example, delay might be a function of four param- d p = 0.89 - 17.75 LN ( PCROSS )
eters [(P(Y_ENC), P(GO|Y), P(CG_ENC), and P(GO|CG)],
where
but assumptions need to be made on some variables in order
dp = average pedestrian delay (s)
to produce a visual plot of others. In application of the mod-
PCROSS = the natural logarithm of the probability of cross-
els, analysts should always use the equation form of the model.
ing [= P(Y_ENC) P(GO|Y) + P(CG_ENC)
For additional details on model development, the reader is
P(GO|CG)].
directed to Appendix K.
The suggested delay model for channelized turn lanes pre-
dicts pedestrian delay as a function of the natural logarithm
Channelized Turn Lane Delay Model
of PCROSS, which is calculated from the four individual proba-
A total of 30 participants (16 pretest and 14 posttest) were bility parameters. The overall model and the PCROSS parameter
included in the analysis. Each observation represents the aver- are significant (p < 0.0001). The adjusted R-square value sug-
age of 12 to 20 trials (6 to 10 round trips with two crossing tri- gests that 79.3% of the variability in the delay is explained by
als each). With the distinction of the two studied crosswalks as the model, which is very high given that inter-participant
well as pretest and posttest observations, the dataset thus con- variability of crossing performance was very high. Figure 25
tains 60 observations. One observation had to be excluded shows the fit of the model against field-observed data. Since
from the dataset since the participant did not encounter any the LN(PCROSS) term represents a combination of encounter
yielding from drivers. As a result the final dataset contained and utilization parameters, it can be used to test the sensitiv-
59 observations. ity of the different probability components. The two curves
Various model forms were tested and are discussed in detail contained in Figure 25 therefore show the predicted delay for
in Appendix K. Model selection was guided by statistical sig- 50% opportunity utilization (both crossable gaps and yields)
nificance (overall model significance, parameter significance, and 100% utilization. The latter approximates the delay a
and adjusted R-square value), as well as practical significance sighted pedestrian may have experienced if encountering the
(model simplicity, reasonableness of results, fit with field- same crossing opportunities. The curves were created by vary-
observed data). Equation 1 shows the suggested pedestrian ing P(Y_ENC) and P(CG_ENC) from 0 to 1.0 while keeping
delay model. the values of P(GO|Y) and P(GO|CG) constant at 0.5 and 1.0.
Channelized Turn Lane Delay Model
100
50% Utilization
90
Pedestrian Delay and Min Delay (s).
100% Utilization
80
Raw Data- Pre
70
Raw Data-Post
60 Raw Data Min Delay
50
40
30
20
10
0
0% 20% 40% 60% 80% 100%
P(Y_ENC)=P(CG_ENC)
This figure shows a chart of the developed mixed-priority delay model for the CTL. The chart plots
the relationship between the probability of encountering yield and gap events on the x-axis and
the pedestrian all crossing opportunities and pedestrians who only utilize 50% of opportunities.
The graph further shows the field-observed data points for pretest and posttest, as well as the
field-observed (theoretical) minimum delay for the pedestrians.
Figure 25. Graphical comparison of CTL delay model against field data.
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The two curves are plotted against the raw average delay data Equation 2. Suggested pedestrian delay model for single-
for pretest and posttest conditions. Finally, the figure contains lane roundabouts.
the theoretical average minimum delay for each data point.
In interpreting the figure, the raw data should be compared d p = 0.78 - 14.99 LN ( PCROSS )
against the 50% utilization curve, while the minimum delay
where
data should be compared against the 100% utilization curve.
dp = average pedestrian delay (s)
The figure shows that the general trends of the model delay
PCROSS = probability of crossing [= P(Y_ENC) P(GO|Y) +
curves fall within the area of observed data, as was suggested
P(CG_ENC) P(GO|CG)].
by the high model adjusted R-square value. The exponential
model form predicts high delays when PCROSS is in the range The suggested delay model for channelized turn lanes pre-
of 0 to 20%, corresponding to a very low occurrence of cross- dicts pedestrian delay as a function of the natural logarithm
ing opportunities (since utilization is fixed). As the availabil- of PCROSS, which is calculated from the four individual proba-
ity of crossing opportunities increases, the delay drops, which bility parameters. The overall model and the PCROSS parame-
is supported by the field data. The distinction between pretest ter are significant (p < 0.0001). The adjusted R-square value
(filled circles) and posttest (hollow circles) shows a general suggests that 63.6% of the variability in the data is explained
trend toward higher PCROSS and lower delay after treatment by the model, which is very high given that inter-participant
installation. In this context it is important to emphasize that variability of crossing performance was very high.
the treatment effect is not explicitly included in the model. Figure 26 shows the fit of the model against field-observed
While this was tried in model development, the treatment data. Since the LN(PCROSS) term represents a combination of
dummy variable was not significant with the PCROSS variable encounter and utilization parameters, it can be used to test
also in the model. This indicates that any treatment effect is the sensitivity of the different probability components. The
implicitly represented in the variability of PCROSS. This finding two curves contained in Figure 26 therefore show the pre-
gives confidence to the model form and allows its application dicted delay for 50% opportunity utilization (both crossable
beyond the treatments tested by varying the underlying gaps and yields) and 100% utilization. The latter approxi-
probability terms. One example for this type of sensitivity mates the delay a sighted pedestrian might have experienced
analysis is the 100% utilization curve that hypothesizes the if encountering the same crossing opportunities. The curves
delay experienced by a (sighted) pedestrian who utilizes were created by varying P(Y_ENC) and P(CG_ENC) from 0
every opportunity. The trend for that curve fits well with the to 1.0 while keeping the values of P(GO|Y) and P(GO|CG)
observed minimum delay raw data, which represents the constant at 0.5 and 1.0. The two curves are plotted against the
minimum theoretical delay if the very first crossing opportu- raw average delay data for pretest and posttest conditions.
nity was always utilized by a participant. Finally, the figure contains the theoretical average minimum
delay for each data point. In the interpretation of the figure,
the raw data should be compared against the 50% utilization
Single-Lane Roundabout Delay Model
curve, while the minimum delay data should be compared
A total of 40 participants were included in the analysis from against the 100% utilization curve.
three different single-lane roundabout sites. Each observation The figure shows that the general trends of the model delay
represents the average of four or more crossing trials at a par- curves fall within the area of observed data, as was suggested
ticular site. With the distinction of entry versus exit crossings, by the high model adjusted R-square value. The exponential
the dataset contained 80 observations. However, four obser- model form predicts high delays when PCROSS is low, correspon-
vations had to be excluded since these participants had one or ding to a very low occurrence of crossing opportunities (since
more zero observations because they either didn't encounter utilization is fixed). As the availability of crossing opportuni-
any crossable gaps or because no drivers yielded for them. As ties increases, the delay drops, which is supported by the field
a result, the final dataset contained 76 observations. No treat- data (black circles). Similar to the CTL model, the 100% uti-
ments were installed at any of the tested single-lane round- lization curve fits well with the observed minimum delay data.
abouts, and consequently there is no posttreatment data.
Various model forms were tested and are discussed in detail
Two-Lane Roundabout Delay Model
in Appendix K. Model selection was guided by statistical sig-
nificance (overall model significance, parameter significance, The two-lane roundabout model utilized different inde-
and adjusted R-square value) as well as practical significance pendent variables that are consistent with the revised analysis
(model simplicity, reasonableness of results, fit with field- framework for two-lane approaches presented in Chapter 4.
observed data). Equation 2 shows the suggested pedestrian The reader may recall that crossing a two-lane roundabout
delay model. requires the consideration of three different event conditions
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Single-Lane Roundabout Delay Model
100
90 50% Utilization
Pedestrian Delay & Min. Delay (s)
80 100% Utilization
Raw Data - D elay
70
Raw Data - M in. Delay
60
50
40
30
20
10
0
0% 20% 40% 60% 80% 100%
P(Y_ENC)=P(CG_ENC)
This figure shows a chart of the developed mixed-priority delay model for the single-lane roundabout.
The chart plots the relationship between the probability of encountering yield and gap events on the
x-axis and the pedestrian delay in seconds on the y-axis. The graph shows two curves, representing
pedestrians with 100% utilization of all crossing opportunities and pedestrians who only utilize 50%
of opportunities. The graph further shows the field-observed data points for pretest and posttest
as well as the field-observed (theoretical) minimum delay for the pedestrians.
Figure 26. Graphical comparison of single-lane roundabout delay model
against field data.
for each lane (crossable gap, non-crossable gap, and yield) exit (entry), and going back] at each of two approaches of the
resulting in nine different combinations. A crossable situa- two-lane roundabout. However a few of the observations had
tion is defined as encountering either a yield or a crossable to be excluded from the dataset since the participant did not
gap. From these nine combinations, four yield crossable sit- encounter any yields from drivers. As a result the final dataset
uations in both lanes: YieldYield, YieldCG, CGYield, and contained 124 observations
CGCG. There are four other combinations that have cross- Various model forms were tested and are discussed in detail
able situations in only one of the lanes (Yieldnon-CG, in Appendix K. Model selection was guided by statistical sig-
CGnon-CG, non-CGYield, and non-CGCG). The remain- nificance (overall model significance, parameter significance,
ing combination has a non-crossable gap in both lanes. These and adjusted R-square value) as well as practical significance
combinations introduce new probability terms that are defined (model simplicity, reasonableness of results, fit with field-
below: observed data). Equation 3 shows the suggested pedestrian
delay model.
· PA_Dual: This is the probability of encountering a cross- Equation 3. Suggested pedestrian delay model for two-
able situation (i.e., crossable gap or yield) in both lanes. lane roundabouts.
· PU_Dual: This is the probability of utilizing a situation
that has a crossable situation (crossable gap or yield) in d p = 1.9 - 21.0 LN ( PDual_CROSS )
both lanes.
where
· P_Dual_Cross: This is the probability of crossing when
dp = average pedestrian delay (s)
both lanes have a crossable situation, defined as PA_Dual
PDual_Cross = probability of crossing when both lanes have a
times PU_Dual.
crossable situation in the form of a crossable
gap or a yield (= PA_Dual PA_Dual).
A total of 31 participants were included in the analysis,
including pretest (18 participants) and posttest treatment The suggested delay model for channelized turn lanes pre-
(13). Each observation represents the average of 16 crossing dicts pedestrian delay as a function of the natural logarithm
trials [four round trips, each with crossing trials from curb to of PDUAL_CROSS, which is calculated from the availability and
splitter island at entry (exit), from splitter island to curb at encounter probability of dual crossing opportunities. The
